Theorem imaeq2 5097

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 5033	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4986	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4762	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4762	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2290	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1398   ran crn 4750   |` cres 4751   "cima 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762

This theorem is referenced by:  imaeq2i  5099  imaeq2d  5101  fimadmfo  5599  ssimaex  5738  ssimaexg  5739  isoselem  5993  f1opw2  6261  supp0cosupp0fn  6467  fopwdom  7089  ssenen  7105  fiintim  7191  fidcenumlemrk  7224  fidcenumlemr  7225  sbthlem2  7228  isbth  7237  ennnfonelemp1  13157  ennnfonelemnn0  13173  ctinfomlemom  13178  ctinfom  13179  tgcn  15073  iscnp4  15083  cnpnei  15084  cnima  15085  cnconst2  15098  cnrest2  15101  cnptoprest  15104  txcnp  15136  txcnmpt  15138  metcnp3  15376